Conservation of Number Piaget

12 Key excerpts on "Conservation of Number Piaget"

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  Developing Cognitive Competence

  New Approaches To Process Modeling

  • Tony J. Simon, Graeme S. Halford, Tony J. Simon, Graeme S. Halford(Authors)
  • 2015(Publication Date)
  • Psychology Press

    (Publisher)

  A central tenet of Piagetian theory (Piaget, 1952, 1970) is that the acquisition of conservation knowledge is a crucial step in the child's development of mature conceptual capabilities. Piaget (1968, p.978) defined conservation as follows:

  We call "conservation" (and this is generally accepted) the invariance of a characteristic despite transformations of the object or of a collection of objects possessing this characteristic. Concerning number, a collection of objects "conserves" its number when the shape or disposition of the collection is modified, or when it is partitioned into subsets.

  As we stated, children's knowledge about the effects of transformations must be empirically derived in the first instance because all transformations have different effects on different physical dimensions of the transformed material. For example, whether or not the pouring transformation conserves quantity depends on what is poured and what is measured:

  If we pour a little sugar into red sugar water, we do not change temperature, amount, height, width, or redness, but we increase sweetness. If we add more of an identical concentration, we do not change temperature, redness or sweetness; however the amount increases, as does liquid height, but not width (in a rigid container). On the other hand, if we add water, we increase two extensive quantities (amount, liquid height), reduce two intensive quantities (redness, sweetness), and leave one unchanged (temperature).

  (Klahr, 1982, pp. 68-69)

  Therefore, a central component of what must be learned, either in training studies or by being naturally acquired by the child outside the laboratory, are the linkages between transformational attributes and their dimensional effects as measured in a variety of contexts.

  The centrality of conservation concepts to most theories of cognitive development produced a vast database of empirical results. Nevertheless, a computational model that can account for the regularities has yet to be fully specified. There are structural and processing accounts of the knowledge used by a child who "has" conservation, as well as global characterizations of the acquisition of that knowledge, such as Piaget's assimilation and accommodation processes, Klahr and Wallace's (1976) time-line processing, and Halford's (1982) levels of cognitive systems. However, neither these nor any other accounts completely stated a set of operations and their interaction with a specified learning mechanism and shown this to produce the pattern of behavior observed in children acquiring conservation knowledge.

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  Cognitive Development

  An Information-Processing View

  • David Klahr, J. G. Wallace(Authors)
  • 2022(Publication Date)
  • Routledge

    (Publisher)

  5 Conservation of Quantity

  The classic version of the Piagetian test for conservation of quantity starts with the presentation of two distinct collections of equal amounts of material (for example, two rows of beads, two vessels of liquid, or two lumps of clay). First the child is encouraged to establish their quantitative equality (for example, “Is there as much to drink in this one as in that one?” or “Is it fair to give this bunch to you and that bunch to me?”). Then he observes one of the collections undergo a transformation that changes some of its perceptual features while maintaining its quantity (for example, stretching, compressing, pouring into a vessel of different dimensions). Finally, the child is asked to judge the relative quantity of the two collections after the transformation. To be classified as “having conservation” the child must be able to assert the continuing quantitative equality of the two collections without resorting to a requantification and comparison after the transformation; that is, his response must be based not upon another direct observation, but rather upon recognition of the “logical necessity” for initially equal amounts to remain equal under “mere” perceptual transformations.

  The problem for students of cognitive development has been stated by Wallach (1969 ):

  Much as conservation is later taken entirely for granted, Piaget... and others... have shown clearly that until the age of six or seven children believe that quantities do change under such transformations. What happens at that point? How do these children come, like us, to consider it an absurdity even to ask whether amounts might change with different containers or arrangements? [p. 191]

  This chapter is organized as follows: first, we present a theory of concrete operational performance on tasks involving conservation of discontinuous quantities such as collections of beads. Next, we consider the developmental course of the system from the preoperational to the concrete operational level of performance with discontinuous quantity; then we describe the extension of this capability to conservation of continuous quantities such as lumps of plasticene or jars of water. In both sections the theories which will be advanced are predicated on the account of the three quantification operators and their developmental interrelationship presented in Chapter 3

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  Learning and the Development of Cognition (Psychology Revivals)

  • Barbel Inhelder, Hermine Sinclair, Magali Bovet(Authors)
  • 2014(Publication Date)
  • Taylor & Francis

    (Publisher)

  The epistemological interest of the development of conservation concepts is evident. These concepts are neither preformed in the child nor acquired by means of simple observation of real events, but are the product of a process of elaboration which Piaget seeks to explain in terms of equilibration and autoregulation. Conservation concepts are also of special interest to psychologists, because their growth is governed by very regular laws of development. Finally, psychopathological studies of retardation or deviation of normal development highlight the importance of conservation concepts from a different perspective.

  Research into the development of concepts of conservation of quantity has been followed with considerable interest by both child psychologists and developmental psychologists and a number of replication studies have been carried out. In the original conservation studies a given quantity of liquid was poured into glasses of different sizes (Piaget and Szeminska, 1941) or a ball of modeling clay was first changed in shape and then broken into several smaller pieces (Piaget and Inhelder, 1941). These studies revealed that the child’s initial understanding of conservation is based on a general undifferentiated concept of invariance which provides the basis for subsequent, more specific quantifications and measurements (e.g., of height and length). This first notion of conservation of continuous (or physical) quantity is developed before any actual physical quantification of mass, volume, or weight is possible. Differentiation between, on the one hand, the underlying synchronic operation structures and, on the other, the continuous and causal action is only partial (Piaget and Garcia, 1971).

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  Emerging Perspectives on Gesture and Embodiment in Mathematics

  • Laurie D. Edwards, Francesca Ferrara, Deborah Moore-Russo(Authors)
  • 2017(Publication Date)
  • Information Age Publishing

    (Publisher)

  A full understanding of conservation requires understanding that, al- though a transformation alters appearance, it does not in fact alter quan- tity. A child who conserves understands that, despite the misleading appear- ance of the transformed object, the quantity has not changed. In contrast, children who do not yet conserve are swayed by the perceptual appear- ance of a transformed quantity. Conserving children’s responses reveal that they use a variety of operations basic to mathematical understanding (including one-to-one correspondence, counting, reversibility, identity, and quantitative comparison) to justify why quantities remain the same after Figure 2.1 Schematic of conservation task. Embodied Knowledge in the Development of Conservation of Quantity  29 transformation (e.g., Rains, Kelly, & Durham, 2008). Understanding that quantity is conserved under certain sorts of transformations can be char- acterized in terms of progress from a perceptually based approach to an abstract, logic-based, or “cognitive” approach to understanding quantity (e.g., Piaget, 1941/1952; Schultz, 1998). A great deal of research on children’s understanding of conservation has focused on patterns of change over development. Piaget viewed children’s performance on conservation tasks as indicative of their stage of cogni- tive development and consequently focused on discontinuities or sudden shifts in children’s performance (Piaget, 1941/1952). Other investigators have challenged Piaget’s views, criticizing on numerous grounds his no- tion of “stage” and his claim that stage-related cognitive processes constrain children’s learning.

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  Critical Readings on Piaget

  • Leslie Smith(Author)
  • 2002(Publication Date)
  • Routledge

    (Publisher)

  Piaget (1952) viewed number development as an integral part of the development of logical reasoning, and that development as a product of very general properties of the child’s interactions with the world. He and the many researchers who followed his lead found dramatic differences in the numerical reasoning of preschoolers versus older children, which they attributed to the dependence of a concept of number on concrete-operational structures that are not attained until around 6 or 7 years of age. Recently, however, interest has grown in a very different characterization of numerical development, one that emphasizes the primacy of representational uses of number, such as counting, rather than logical reasoning, and that correspondingly views young children as much more competent than Piaget believed (e.g. Gelman and Gallistel 1978; Wynn 1992).

  This alternative conceptualization, and the research supporting it, has been heavily influenced by Chomsky’s (1957) distinction between competence and performance, which inspired work in two directions. First, it led to the recognition that children may have a hypothesized concept or cognitive ability yet perform poorly on cognitive tasks assessing that ability for a variety of reasons. Thus, many researchers have argued that young children in fact understand number conservation but fail the standard Piagetian conservation problem because of linguistic difficulties, misleading social-interactional cues, or other performance factors (e.g. Bryant 1972; Gelman 1972; McGarrigle and Donaldson 1975; Rose and Blank 1974). Second, it led to the recognition that children may have implicit knowledge which is evident in regularities in their behavior even if they cannot express that knowledge explicitly (Gelman and Gallistel 1978). This insight led to extensive study of children’s counting as a behavioral system in which implicit numerical knowledge might be revealed. Moreover, building on the idea that early counting is already conceptually based is the idea that laterdeveloping numerical reasoning abilities, like conservation, are attained by gaining explicit access to the conceptual knowledge that was originally implicit in their counting: “Later developing number concepts often involve the accessing of implicit knowledge embedded in the structures which characterize early number concepts” (Gelman 1982, p. 217).

  There are two potential difficulties for this characterization of number development, however—one pertaining to the general claim of early numerical competence and the other to the assertion that counting plays an important role in the development of conservation and related forms of numerical reasoning. The first stems from the double-edged nature of the competence-performance distinction: that is, if no task can directly tap children’s underlying competencies, then data based on children’s performance may overestimate as well as underestimate children’s knowledge. Some evidence for the occurrence of such “false positives” comes from recent studies in which procedural modifications that were designed to increase conservation also increased the occurrence of conservation-like responses when they were not appropriate (e.g. Light and Gilmour 1983). Clearly, the conclusion that conservation and related forms of numerical reasoning develop much earlier than previously thought needs to be reevaluated in the light of this kind of evidence. The second difficulty stems from evidence of limitations on children’s early understanding of counting, particularly their understanding of how counting can be used to compare two sets (Saxe 1977; Sophian 1987, 1988). It is precisely this use of counting that is most relevant to conservation problems. Thus this evidence seems to undermine proposals that young preschoolers attain conservation and related forms of reasoning on the basis of their proficiency in counting.

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  Routledge Library Editions: Education Mini-Set O Teaching and Learning 14 vols

  • Various(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  Even though he knows there was originally the same quantity in each glass, his inability to de-centre or handle two relationships at one time does not permit him to take into account features which could compensate for the distorting effects. It looks more, therefore it is more. 124/ Aspects of Learning Another weakness in the child's thinking is also revealed. He is unable to see the possibility of returning the liquid to its original state; he is unable to grasp the problem as a whole or to range his thoughts backwards to the original starting point. His thinking is irreversible, and this lack of reversibility is closely linked to another important aspect, that of conservation. By conservation Piaget means the invariance of quantities, that is, no matter what transformations take place in shape it is always possible to return to the original. In the example above the liquid could be poured back into the original container. This idea of conservation is also illustrated by taking a ball of plasticine. It is shown to a child of about 5 or 6 years, and then it is rolled ant into a long, thin, worm-like shape and the child asked if there is more or less plasticine than before. Until he conserves in his mind the original quantity of plasticine the child wiIl still be governed by his perceptions. It is not until conservation is acquired at approxi- mately the age of 7 years that the child moves into the first period of logical thought - the concrete operational period. THE CONCRETE OPERATIONAL PERIOD (Seven to Eleven Years) This is really the first reasoning stage. The word operational is Piaget's term for what we would probably call logical; he means that the child is capable of thinking over actions which previously he had carried out overtly, and operations are actions which can be carried out in thought and are reversible.

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  Learning Mathematics

  Issues, Theory and Classroom Practice

  • Anthony Orton(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  Must We Wait until Pupils Are Ready? 61 Piaget set himself the task of finding out . . . how the principles of conservation and of reversibility, as applied to numbers and to spatial thinking, develop in the minds of young children. The two principles are fundamental to all mathematical (and logical) thinking. . . . understanding cannot be taught nor does it come by itself, independently of experience . . . This does not mean that there is nothing the teacher can do except wait for the dawn of understanding. He can provide the kind of experience which will assist the child to move from intuitive to operational thinking. Children learn mathematical concepts more slowly than we realized. They learn by their own activities. Although children think and reason in different ways they all pass through certain stages depending on their chronological and mental ages and their experience. The report of the Mathematical Association (1970) on primary mathematics included an appendix on 'Understanding and mathematics' which incorporated much of the spirit of Piaget within a broader review of what was known about learning. Caution was recommended, however, in the application of any interpretation of Piagetian theory, as is shown in these two extracts (p. 153): Although these stages . . . have been broadly substantiated by a large number of research workers, we should show due caution in accepting them as a permanent feature in childhood development; . . . it is important not to discourage experiment, in the belief that what has been found is an unalterable feature of childhood development. We have only to compare the thinking of primitive adults with that of educated children in industrial societies to see the vast changes which are possible. Piagetian views provided the underlying rationale for the book by Lovell (1971b) prepared as a guide for teachers of young children.

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  Teaching and Learning Mathematics

  A Teacher's Guide to Recent Research and Its Application

  • Marilyn Nickson(Author)
  • 2004(Publication Date)
  • Continuum

    (Publisher)

  Number and counting skills The studies by Hughes (1986) following on work by Donaldson (1978) were foremost in the UK in challenging the results of some of Piaget's work, particularly with respect to children's learning of number skills. Hughes (1986) found that children's 'counting strategies are frequently untaught, and are meaningful attempts by the child to solve the problems confronting them' (p. 35). He also found that 'even pre-school children are able to represent small quantities, either spontaneously, or with small amounts of prompting' and it should not be presumed that they have necessarily been trained by parents in meaningless parroting (ibid., p. 77). Although forms of representation mainly take the form of one-to-one correspondence and are pictographic or iconic, they are still abstractions of a physical counting situation. Number and Calculation 11 Gray et al. (1997) use the activity and representation of counting in their develop-ment of a cognitive theory related to children's conceptual development in mathematics, and devised the notion of procept to take into account the role of mathematical symbolization where it can represent a process (to do something) or a concept (to know something). They approach this through a consideration of how children learn to count and to use the names of numbers (see below). They note that learning to count is grounded in actions related to physical objects, but the physical objects themselves essentially need to be ignored: it is the actions carried out by the child that are important and which have to create an 'object of the mind' (ibid. , p. 115): For some there may be a cognitive shift from concrete to abstract in which the concept of number becomes conceived as a construct that can be manipulated in the mind.

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  Experimental Psychology Its Scope and Method: Volume VII

  Intelligence

  • Pierre Oléron, Jean Piaget, Bärbel Inhelder, Pierre Gréco(Authors)
  • 2014(Publication Date)
  • Taylor & Francis

    (Publisher)

  From the psychological point of view, the two main difficulties which stand in the way of this solution are that, as we have seen, elementary intuitions of number are not immediately numerical but only ‘prenumerical’ for want of additivity and particularly of conservation. Also the transition from these preoperational structures to the operational concept of number proceeds according to stages that are surprisingly parallel (with approximate term-for-term synchronization) to those which we indicated in the construction of groupings of classes and of relations. These two affirmations (initial non-conservation of numerical sets and parallel succession of stages) can be controlled by a very simple experiment. This has been repeated by several researchers (Churchill, Laurendeau and Pinard, etc.). It was devised by one of us (Piaget) in 1919–1920, to distinguish between normal children and young epileptics in the Voisin Division of La Salpêtrière (in fact normal children reacted in exactly the same way between the ages of 4 and 6, as was later observed with A. Szeminska!) 1 The child is first shown a series of 6–7 blue counters with a small space between them and then given a box of red counters. He is asked to put on the table as many red counters as there are blue. Four stages can be observed. During the first, the child merely constructs with the red counters a row of the same length as the row of blue counters. He judges quantity by the space that is filled. During the second stage, the subject establishes a term-for-term but optical correspondence (each red opposite a blue). When this optical configuration is destroyed by spacing out one of the rows, the child thinks that there is no longer equivalence either in quantity or in number

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  Child Psychology

  Development in a Changing Society

  • Robin Harwood, Scott A. Miller, Ross Vasta(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  In the first example in Figure 7.5, the conservation of number problem, children view two rows of five chips (Piaget & Szeminska, 1952). As long as the chips are arranged in one-to-one correspondence, even a 3- or 4-year-old can tell us that the two rows have the same number of objects. But if, while the child watches, you spread out one of the rows so that it is longer than the other, virtually every 3- and 4-year old will say that the longer row now has more chips. Asked why, the child finds the answer obvious—because it is longer. In Piaget’s terms, the child’s attention is centered on the length of the row, leading to the failure to conserve the number. Centration, then, is a perceptually biased form of responding characteristic of young children. For the preschooler, what seems critical is how things look at the moment. The child’s attention is captured by the most salient, or noticeable, element of the display, which in the number task is the length of the rows. Once his or her attention has been captured, the child finds it difficult to take account of other information—for example, the fact that the rows differ not only in length but also in density. Thus the child is easily fooled by appearance and often, as in the conservation task, arrives at the wrong answer. Focus on Appearances Centration also leads to another striking limitation in preoper- ational children’s thought—the inability to dinstinguish appearance from reality. A striking example is found in Rhetta DeVries’s “Maynard the Cat” study (1969). De- Vries first introduced preschoolers to a friendly black cat named Maynard. She then transformed Maynard by putting a realistic and fierce-looking dog mask on him.

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  Culture and Cognition

  Readings in Cross-Cultural Psychology

  • J. W. Berry, P. R. Dasen(Authors)
  • 2019(Publication Date)
  • Taylor & Francis

    (Publisher)

  ignoring the perceptual differences, but of his coordinating them by means of a logical understanding of the transformation. This is the same criticism that we made of Bruner and Greenfield’s method of masking the disturbing perceptual features in order to obtain correct conservation responses.

  These remarks explain why we are somewhat reluctant to accept as such Furby’s proposed framework for cross-cultural interpretations.

  General Discussion (A)

  (a) Adults

  How do our findings relate to the hypothesis, which directed the choice of the two types of notions studied in the present research, based on the extent of usage in the everyday activities of the population under study?

  With regard to the activities associated with the notions of conservation of quantity, the following remarks may be made. In the area of weight, the information collected on the household activities of the women showed that they often used weight measurement, particularly when baking bread. Scales were never used for this activity, although the women knew how to use such an instrument. The procedure was to estimate the required weight in their hands. It was obvious from the interviews that the women were extraordinarily skilful in equalizing the weight of the two clay balls used in the experiment, and that weighing in the hand was an effective measuring instrument for them.

  The unusual response of a refusal to give a judgement before being able to hold the modified ball of clay in their hands, and the behaviour which consisted of an initial conservation judgement followed by a regression as soon as attention was drawn to the changed dimensions, were observed only in the women. The behaviour of the men consisted of either stable conservation responses, or truly nonconserving ones.

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  Childhood and Adolescence

  Voyages in Development

  • Spencer Rathus(Author)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The demands of the standard conservation task may also present a misleading picture of the child’s knowledge. Piaget and other experimenters filled identical bea-kers with the same amount of water and then poured water from one of the beakers into a beaker of another shape. Before pouring the water, the experimenter typically asked the child whether both beakers have the same amount of water and instructed the child to watch the pouring carefully. In doing so, perhaps the experimenter is “leading the witness”—that is, leading the child to expect a change. 1. According to Piaget, _______ play is based on the use of symbols. 2. _______ are mental acts in which objects are changed or transformed and can then be returned to their original states. 3. Piaget used the three-mountains test to show that preoperational children are _______. 4. The type of thinking in which children attribute will to inanimate objects is termed _______ thinking. 5. In _______ reasoning, children reason from one specific event to another. 6. The law of _______ holds that properties of substances, such as volume, mass, and number, remain the same even when their shape or arrangement changes. 7. Preoperational children focus on _______ (how many?) dimension(s) of a problem at once. Reflect & Relate: Preoperational children focus on one dimension of a problem at a time. Do we as adults sometimes focus on one dimension of a situation at a time? If you injure someone in an accident, should you be held responsible? (Note the two elements: the injury and the fact that it is accidental.) Most people would probably say, “An accident is an accident.” But what if a utility injures 8 million people in a nuclear accident? Should it be held responsible? (That is, does the enormity of the damage affect responsibility?) Section Review Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

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